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Problem 12.R.65
65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.
A hyperbola with vertices (0, ±2) and directrices y = ±1
Problem 12.R.56c
53–57. Conic sections
c. Find the eccentricity of the curve.
x²/4 + y²/25 = 1
Problem 12.R.12a
10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = ln t, y = 8ln t², for 1 ≤ t ≤ e²; (1, 16)
Problem 12.R.17
14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.
The circle x ² + y ² =9, generated clockwise
Problem 12.R.4
3–6. Eliminating the parameter Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.
x = sin t - 3, y = cos t + 6; 0 ≤ t ≤ π
Problem 12.R.7a
7–8. Parametric curves and tangent lines
a. Eliminate the parameter to obtain an equation in x and y.
x = 8cos t + 1, y = 8sin t + 2, for 0 ≤ t ≤ 2π; t = π/3
Problem 12.R.58
58–59. Tangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility.
x²/16 - y²/9 = 1; (20/3, -4)
Problem 12.R.1e
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
e. The hyperbola y²/2 - x²/4 = 1 has no x-intercept.
Problem 12.R.46
44–49. Areas of regions Find the area of the following regions.
The region inside the limaçon r=2+cosθ and outside the circle r=2
Problem 12.R.1c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. The polar coordinates (3, -3π/4) and (-3, π/4) describe the same point in the plane.
Problem 12.R.54d
53–57. Conic sections
d. Make an accurate graph of the curve.
x = 16y²
Problem 12.R.26
24–26. Sets in polar coordinates Sketch the following sets of points.
4 ≤ r² ≤ 9
Problem 12.R.36a
Polar conversion Consider the equation r=4/(sinθ+cosθ).
a. Convert the equation to Cartesian coordinates and identify the curve it describes.
Problem 12.R.51
51–52. {Use of Tech} Arc length of polar curves Find the approximate length of the following curves.
The limaçon r=3−6cosθ
Problem 12.R.1a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. A set of parametric equations for a given curve is always unique.
Problem 12.R.31
27–32. Polar curves Graph the following equations.
r = 3 sin 4θ
Problem 12.R.60a
A polar conic section Consider the equation r² = sec2θ
a. Convert the equation to Cartesian coordinates and identify the curve.
Problem 12.R.54a
53–57. Conic sections
a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.
x = 16y²
Problem 12.R.35
Polar conversion Write the equation r ² +r(2sinθ−6cosθ)=0 in Cartesian coordinates and identify the corresponding curve.
Problem 12.R.55c
53–57. Conic sections
c. Find the eccentricity of the curve.
y² - 4x² = 16
Problem 12.R.41b
40–41. {Use of Tech} Slopes of tangent lines
b. Find the slope of the lines tangent to the curve at the origin (when relevant).
r = 1 −sin θ
Problem 12.R.60b
A polar conic section Consider the equation r² = sec2θ
b. Find the vertices, foci, directrices, and eccentricity of the curve."
Problem 12.R.72a
Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.
a. For what value of p is P tangent to H?
Problem 12.R.29
27–32. Polar curves Graph the following equations.
r = 3 cos 3θ
Problem 12.R.53b
53–57. Conic sections
b. Use analytical methods to determine the location of the foci, vertices, and directrices.
x² - y²/2 = 1
Problem 12.R.43
42–43. Intersection points Find the intersection points of the following curves.
r= √(cos3t) and r= √(sin3t)
Problem 12.R.70
Conic parameters: A hyperbola has eccentricity e = 2 and foci (0, ±2). Find the location of the vertices and directrices.
Problem 12.R.56a
53–57. Conic sections
a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.
x²/4 + y²/25 = 1
Problem 12.R.10a
10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = t² + 4, y = -t, for -2 < t < 0; (5, 1)
Problem 12.R.57b
53–57. Conic sections
b. Use analytical methods to determine the location of the foci, vertices, and directrices.
4x² + 8y² = 16